Geometric Momentum: the Proper Momentum for a Free Particle on a Two-dimensional Sphere

Abstract

In Dirac's canonical quantization theory on systems with second-class constraints, the commutators between the position, momentum and Hamiltonian form a set of algebraic relations that are fundamental in construction of both the quantum momentum and the Hamiltonian. For a free particle on a two-dimensional sphere or a spherical top, results show that the well-known canonical momentum p_{θ} breaks one of the relations, while three components of the momentum expressed in the three-dimensional Cartesian system of axes as p_{i} (i=1,2,3) are satisfactory all around. This momentum is not only geometrically invariant but also self-adjoint, and we call it geometric momentum. The nontrivial commutators between p_{i} generate three components of the orbital angular momentum; thus the geometric momentum is fundamental to the angular one. We note that there are five different forms of the geometric momentum proposed in the current literature, but only one of them turns out to be meaningful.

17 pages. Phys. Rev. A. 2011 (to appear)

References in corpus (1)

• Quantum mechanics on curved 2D systems with electric and magnetic fields

Cited by in corpus (28)

• Pauli equation for a charged spin particle on a curved surface in an electric and magnetic field
• Geometric influences of a particle confined to a curved surface embedded in three-dimensional Euclidean space
• Quantum particle confined to a thin-layer volume: Non-uniform convergence toward the curved surface
• Magnon-mediated quantum battery under systematic errors
• Quantum motion on a torus as a submanifold problem in a generalized Dirac's theory of second-class constraints
• Can the canonical quantization be accomplished within the intrinsic geometry?
• Geometric momentum for a particle constrained on a curved hypersurface
• Geometric effects resulting from square and circular confinements for a particle constrained to a space curve
• Hamiltonian for a particle in a magnetic field on a curved surface in orthogonal curvilinear coordinates
• Geometric Potential Resulting from Dirac Quantization
• Particle on the sphere: group-theoretic quantization in the presence of a magnetic monopole
• The centripetal force law and the equation of motion for a particle on a curved hypersurface
• The geometric potential of a double-frequency corrugated surface
• $SU(2)$-particle sigma model: Momentum-space quantization of a particle on the sphere $S^3$
• Generalized Centripetal Force Law and Quantization of Motion Constrained on 2D Surfaces
• On relation between geometric momentum and annihilation operators on a two-dimensional sphere
• Quantum mechanics of fermion confined to a curved surface in Foldy-Wouthuysen representation
• Distribution of xp in some molecular rotational states
• Geometric effects of a quarter of corrugated torus
• Spin Force and Torque in Non-relativistic Dirac Oscillator on a Sphere
• General covariant geometric momentum, gauge potential and a Dirac fermion on a two-dimensional sphere
• No existence of the geometric potential for a Dirac fermion on two-dimensional curved surfaces of revolution
• Higgs algebraic symmetry of screened system in a spherical geometry
• Wave functions of the Hydrogen atom in the momentum representation
• Geometric momentum and angular momentum for charge-monopole system
• Generally covariant geometric momentum and geometric potential for a Dirac fermion on a two-dimensional hypersurface
• The Vector and Canonical Components of the Momentum Operator in 3D Euclidean Space Spanned by General Curvilinear Coordinates
• A self-adjoint decomposition of radial momentum implies that Dirac's introduction of the operator is insightful